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Theorem ssex 2709
Description: The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22. This is one way to express the Axiom of Separation ax-sep 2693 (a.k.a. Subset Axiom).
Hypothesis
Ref Expression
ssex.1 |- B e. V
Assertion
Ref Expression
ssex |- (A (_ B -> A e. V)
Proof of Theorem ssex
StepHypRef Expression
1 df-ss 2043 . 2 |- (A (_ B <-> (A i^i B) = A)
2 ssex.1 . . . 4 |- B e. V
32inex2 2707 . . 3 |- (A i^i B) e. V
4 eleq1 1526 . . 3 |- ((A i^i B) = A -> ((A i^i B) e. V <-> A e. V))
53, 4mpbii 193 . 2 |- ((A i^i B) = A -> A e. V)
61, 5sylbi 199 1 |- (A (_ B -> A e. V)
Colors of variables: wff set class
Syntax hints:   -> wi 3   = wceq 953   e. wcel 955  Vcvv 1802   i^i cin 2036   (_ wss 2037
This theorem is referenced by:  ssexi 2710  ssexg 2711  intex 2719  elpm 4320  mapss 4330  inf3lem7 4591  omex 4599  unbnnt 4611  bndrank 4654  scottex 4688  zorn2lem4 4763  ondomon 4828  elnp 5064  suplem2pr 5134  lbinfm 5995  elcncf 7200  unbenlem 7447  lpval 7684  lmclim 7898  sh 8999
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693
This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-in 2041  df-ss 2043
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